Death-Defying Inclined Planes!

GrrlScientist sends a link to this rather wild stunt from India:

How is it possible? What kind of friction is necessary, and is it any more difficult for the cars to do the stunt than it is for the motorcycles?

Before we do any math, I want to think about the problem qualitatively. Let's tally up the forces acting on the vehicles. First, there's gravity. It points vertically downward, straight toward the center of the earth. Second, there's the normal force. "Normal force" means the force normal (in this context a physics technical term meaning perpendicular) to the surface. It's the force the track exerts on the vehicle. It points, as the name indicates, perpendicular to the track surface. Third, there's friction. It points in the opposite direction of wherever the vehicle would be sliding if there were no friction.

You may be a little suspicious of that phrasing, and you should be. How do we know what direction that is? Well, we could calculate it but we don't have to. Let's take a look at the diagram of the situation, conveniently found on the Wikipedia article on the inclined plane:


The gravitational force downward is mg, and the normal force is N. The terms with sine and cosine are just the gravitational force expressed in components - those two forces are exactly equivalent to the downward vertical mg. We do know from Newton's laws that F = ma, i.e,, the sum of all the forces F is equal to the mass times the acceleration. In the situation in the video, the total acceleration parallel to the track is zero. The motorcycles neither ascend nor descend once they reach their cruising altitude, so to speak. But that doesn't mean there's no acceleration. If there were no acceleration, the vehicles would continue in straight lines, sinking into the track like ghosts. Instead, the riders are essentially orbiting; their acceleration is directed toward the center of the circle. Their speed doesn't change, but their direction does.

From freshman physics, we know the force required to keep an object in uniform circular motion is F = v^2/r, toward the center of the circle of radius r. But looking at the diagram, it's easy to see that N points toward the center of the circle, and so does the parallel component of mg. The frictional force f doesn't, and so even if there were no friction and the track were made of ice, N and mg could produce the required acceleration, keeping the riders in place.

Now I'm going to depart from the diagram just a bit to make the math easier. Instead of decomposing mg into parts with respect to the plane, I'm going to decompose N into vertical and horizontal components with respect to the ground.

Since the cars neither rise nor fall, the vertical component of N has to be equal to the downward force mg:




The only horizontal force is the horizontal component of N, and it has to be equal to the force needed for circular motion:


But we already figured out what N was, so we can substitute:


Now with we can solve for v, the speed necessary to stay orbiting the wall even in the absence of friction:


Because sin/cos = tan, and the m's cancel out. Which is nice - it means that it's equally easy (or hard) for the heavy cars and light motorcycles to do the trick, provided they can actually get up to that speed and provided they're both not too big with respect to the turning radius r.

So how fast do they have to go? It's hard to estimate what r is, but we can guess perhaps 10 meters. The angle is also tough, but let's guess 70 degrees just as an estimate. That gives us a value of about 16.4 m/s, or 36 miles per hour. To my unpracticed eye this seems a bit high, but then I've probably overestimated the angle. A 60 degree incline would mean 29mph, for instance. Either way though, it's pretty clearly doable for motorcycles and cars alike. I don't know what the Indian equivalent of OSHA might think, but the laws of physics are fine with this bit of daring.

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I think you are missing something. I saw something like this at a carnival a while back (no cars, just motorcycles), the angle is 90. They did go fast, but not infinitely fast.

Yep. What's missing is friction. Which increases with the centrifugal force. Were everything frictionless, 90* would be a limit. With friction, it can be not only reached, but exceeded.

Thanks for the explanation - presumably the cars and bikes just need to go faster to get up to cruising altitude (and point themselves in the right direction!).

Russell, um what? How do you figure that friction allows you to exceed 90 degrees? Once you are at 90 the slightest vibration will push you off the wall and your down. Anything above 90 doesn't even need to wait for that. 90 is the absolute limit with friction also.

This is a simpler variation of a circus/acrobatics stunt.
For example, see:
Barnum & Bailey Circus: Motorcycles in Cage
and many of the linked videos.

In those cases, the motorcycles are in a sphere and clearly moving on a 90 degree surface. It's also obvious and clear from the video that whenever they cross to the top half of the sphere they can no longer do full circles that are perpendicular to the earth.

Its essential to add in the "centripetal force". For a rotating coordinate system omega * R squared, which is directed away from R. Yes for over 90 degrees it would work (with a little bit of friction). The apparent acceleration in the frames of the riders is clearly tpwards the wall.

By Omega Centauri (not verified) on 19 Mar 2010 #permalink

Oops, that is Omega squared times R. In any case if the time to go completely around was 2 PI seconds Omega would be 1/second, and the radius was ten meters, that would give 10 meters/second**2, which matches gravity. I think R is probably a bit smaller, but Omega is probably a bit faster than one.

By Omega Centauri (not verified) on 19 Mar 2010 #permalink

joshua think of that theme park ride where people are stuck against the inside of a rotating wall. imagine that wall angled more than 90 degrees (but the rotation stays on the same plane), it's conceivable that they could stay stuck against the wall by the vertical component of the normal force which increases as the centrifugal force increases. If the rotation is fast enough that the magnitude of that vertical component exceeds that of the gravitational force then they could stay stuck.

(they would have to start rotating at 90 degrees before increasing the angle and rotation speed of course)

The analysis above assumes the car/motorcycle is a point object. It is actually fairly large relative to the radius so various parts of it are "orbitting" at different radii. Even if we treat the motorcycle and rider as a rigid body then gravity can be said to act through the centre of gravity which is determined solely by the mass distribution, while the acceleration will "act" through a point determined by the mass distribution and the spacial distribution of that mass - for an elongated rigid body those points do not usually coincide. Also, the normal force and friction are actually several forces acting separately on each of the tires and do not point through either the center of gravity or whatever point the Centripetal "force" apears to be acting...
Just to add to the complexity here, the wheels of the motorcycle are spinning masses with angular momentum - and the direction of that is changing as the machine circles the track. So, as well as adding up to provide the "orbital" equilibrium, the forces have to add to the exact torque on the rigid body to provide the correct precession of the wheels AND keep the motorcycle "upright"(relative to the track). Watch the video and it is clear the motorcycles definitly lean towards the top of the track as they orbit.

There not actually be a solution with absolutely no friction due to the need to satisfy both the orbital and "upright" equilibrium requirements, though "lean" as well as speed may provide enough degrees of freedom to satisfy both conditions without friction. As a bike rider, I certainly miss friction when hitting ice patches!

Set this problem up for a computer to solve and you are talking major effort - Give a human a bike and somehow the brain finds the required solution of speed, lean and steering angle, and then has the audacity to stand up and wave!

You can analyze it in a rotating frame (where net F = 0) or an inertial one (where net F = mv^2/r), just don't jump between the two frames within the same solution.

An observation about the "experiment": notice that the cars are going around nose high, so the drive wheels are providing an upward force against gravity. That reduces the speed needed to be in equilibrium in the y direction relative to the "ground" frame Matt used. The orientation of the car is sort of like the way cars race on dirt (most people may only have seen this in "Cars").

You can't go much above 90 deg if you are traveling in a horizontal circle, unless you have sufficient aerodynamic downforce on the car. With aero forces, you can do almost anything.

But it is also "easy" to do almost anything if you are in a vertical circle (loop-the-loop roller coaster) or have some component of a vertical circle (circus example or a Mercedes AMG ad where they took a helical path through a highway tunnel). Check this out ...

That stunt was duplicated by BBC's "Top Gear" in a Renault inside of a large concrete pipe (also on YouTube).

By CCPhysicist (not verified) on 20 Mar 2010 #permalink

I love this video. I used it and similar ones when teaching community college students. Really helps bring things to life

While it's true that there is a speed where the cycles will stay up even without friction, it's also true that because there is friction, the cycles can still stay up while going slower than that speed.

In that case, however, the cars would need more friction than the cycles to stay up at the same speed, or need to go faster than the cycles. However, in the movie, we see them go at the same speed, or even a little slower than the bikes.

Was it just me, or were the front ends of the cars consistently higher up on the wall then their rear ends? It seemed the cars were not actually moving in the direction they were pointed at. I also thought I saw the skid-marks to confirm this. If so, then the drive force at that angle may provide the extra vertical force to keep the cars up.

What concerns me most is the forces applied to that flimsy looking wooden structure.

By Benton Jackson (not verified) on 21 Mar 2010 #permalink

I randomly came across a reference to the same stunt in an obscure movie:
which copies the same stunt in an Elvis Presley movie from 1964. A search on "wall of death" seems to bring up a bunch videos & building demonstrations for the same stunt.

My instincts say that, with friction, it's possible to orbit entirely above the 90 degree point. In the absence of friction, such an orbit, if fast enough, would slide down to 90 degrees. AND, I think it would make a great science demonstration.

It would make a practical way to reduce the space requirements for freeway interchanges, but you'd want to avoid traffic jams.

By Carl Brannen (not verified) on 22 Mar 2010 #permalink

But that doesn't mean there's no acceleration. If there were no acceleration, the vehicles would continue in straight lines, sinking into the track like ghosts

WOW!! The people just stand there with a little guardrail protecting them. I've seen some things similar with motorcycles in a round ball but never an "open pit". Looks really cool but still needs a little more protection.

"I don't know what the Indian equivalent of OSHA might think, but the laws of physics are fine with this bit of daring."

HAHAHAHA best part I was just thinking this is something we'd never see in America...